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arXiv 1808.02018math.CO
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关于完全二部图的公平选择性的一个注记

A Note on the Equitable Choosability of Complete Bipartite Graphs

Jeffrey A. Mudrock, Madelynn Chase, Isaac Kadera, Ezekiel Thornburgh, Tim Wagstrom

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9 pages
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摘要：

2003年，Kostochka、Pelsmajer和West提出了一个类似于公平着色的列表，称为公平可选择性。图$G$的$k$分配$L$将$k$可用颜色的列表$L(v)$分配给每个$v \in V(G)$，并且$G$的公平$L$着色是$G$的适当着色$f$，使得每个$v \in V(G)$和$f$的每个颜色类的$f(v) \in L(v)$具有至多$\lceil |V(G)|/k \rceil$的大小.图$G$被称为是公平$k$ -可选择的，如果当$L$是$G$的$k$ -赋值时，存在$G$的公平$L$ -着色.本文研究完全二部图的公平可选性。Kostochka、Pelsmajer和West的结果表明，如果$k \geq \max \{n,m\}$提供了$K_{n,m} \neq K_{2l+1, 2l+1}$，则$K_{n,m}$是公平的$k$可选择的。我们证明了$K_{n,m}$是公平$k$ -可选择的，如果给出$K_{n,m}$的$m \leq \left\lceil (m+n)/k \right \rceil(k-n)$对于满足$k < \max \{n,m\}$的某些$k$是公平$k$ -可选择的.我们还给出了完全二部图的公平可选择性的一个完整的特征，有一个部集的大小至多为2。

英文摘要：

In 2003 Kostochka, Pelsmajer, and West introduced a list analogue of equitable coloring called equitable choosability. A $k$-assignment, $L$, for a graph $G$ assigns a list, $L(v)$, of $k$ available colors to each $v \in V(G)$, and an equitable $L$-coloring of $G$ is a proper coloring, $f$, of $G$ such that $f(v) \in L(v)$ for each $v \in V(G)$ and each color class of $f$ has size at most $\lceil |V(G)|/k \rceil$. Graph $G$ is said to be equitably $k$-choosable if an equitable $L$-coloring of $G$ exists whenever $L$ is a $k$-assignment for $G$. In this note we study the equitable choosability of complete bipartite graphs. A result of Kostochka, Pelsmajer, and West implies $K_{n,m}$ is equitably $k$-choosable if $k \geq \max \{n,m\}$ provided $K_{n,m} \neq K_{2l+1, 2l+1}$. We prove $K_{n,m}$ is equitably $k$-choosable if $m \leq \left\lceil (m+n)/k \right \rceil(k-n)$ which gives $K_{n,m}$ is equitably $k$-choosable for certain $k$ satisfying $k < \max \{n,m\}$. We also give a complete characterization of the equitable choosability of complete bipartite graphs that have a partite set of size at most 2.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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